Review of Basic Math for Computer Graphics. CENG 315 Computer Graphics M. Abdullah Bulbul

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1 Review of Basic Math for Computer Graphics CENG 315 Computer Graphics M. Abdullah Bulbul

2 Sets m S n S x T y T z T m n set S x y z set T A x B: all possible pairs (a,b) where a A, b B A 2 = S, S S x T = { (m,x), (m,y), (m,z), (n,x), (n,y), (n,z) } S 2 = { (m,m), (m,n), (n,m), (n,n) }

3 Useful sets

4 Trigonometry Radians Angle = length of the unit circle that is cut by the two direcoons. Angles degrees = radians * 180/π

5 Pythagorean theorem Right triangle sin(x) =? cos(x) =? tan(x) =? cot(x) =? csc(x) = 1/sin(x) sec(x) = 1/cos(x) Unit circle

6 Useful identities sin(-x) = -sin(x) cos(-x) = cos(x) tan(-x) = -tan(x) sin(π/2 - x) = cos(x) cos(π/2 - x) = sin(x) tan(π/2 - x) = cot(x)

7 Useful identities Pythagorean idenooes: sin 2 (x) + cos 2 (x) =? sec 2 (x) tan 2 (x) =? sin(a+b) = sin(a)cos(b) + sin(b)cos(a) sin(2a) = 2sin(a)cos(a) sin(a-b) = sin(a)cos(b) - sin(b)cos(a) cos(a+b) = cos(a)cos(b) sin(a)sin(b) cos(2a) = cos 2 (a) sin 2 (a) cos(a-b) = cos(a)cos(b) + sin(a)sin(b)

$Vectors A vector: a length & a direcoon a Usually wri\en as a (bold) Absolute posioon is not$

8 Vectors A vector: a length & a direcoon a Usually wri\en as a (bold) Absolute posioon is not important Magnitude = a Used to store offsets, displacements, locaoons To define a locaoon you need an origin

9 Vector addition a a + b a b b b + a commutaove: a+b = b+a

10 Unary minus and subtraction a -a Check subtracoon on a parallelogram

11 2D Cartesian Space a = x a y a a T = x a y a a = 4x 3y

12 2D Cartesian Space A 2D vector can be wri\en as a combinaoon of any two nonzero vectors that are not parallel Linear independence c = k 1 a + k 2 b Exercise

13 Linear independence

14 Linear independence & basis Any two parallels that are not parallel forms a 2D basis. Can go any where by a combinaoon of them Unit vectors are special case Orthonormal basis Simplifies things

15 Vector Multiplication Dot (scalar) product Cross product

16 Dot (scalar) product b x a a.b = b.a =?

17 Dot (scalar) product b x a.b = b.a =? = a b cos(x) a DistribuOve: a.(b+c) = a.b + a.c x = cos -1 ( a.b / ( a b ) ) (most common use)

18 Dot product in cartesian coordinates x a y a x b y b a.b =. =?

19 Dot product in cartesian coordinates x a y a x b y b a.b =. =? (Fairly simple) = x a x b + y a y b

20 Common uses of dot product Find angle between two vectors (light and surface example) ProjecOon of a vector on another one Computed easily on cartesian components

21 Projection of a vector on another b a b x b a a

22 Projection of a vector on another b a b x b a a infer it (5 minutes)

23 Cross product The result of a dot product was scalar The result of a cross product is a vector Perpendicular to both vectors Length: Area of the parallelogram a b = a b sin(ϕ) DirecOon: Right hand rule

24 Cross product X Y = +Z Y X = -Z Y Z = +X Z Y = -X Z X = +Y X Z = -Y a b = -b a a a =? a (b+c) = a b + a c Be\er on the board

25 Cross product on Cartesian space x y z a b = a x a y a z = b x b y b z a y b z a z b y a z b x a x b z a x b y a y b x

26 Orthonormal bases Managing coordinate systems is important! Orthonormal bases is the key. In 2D: u and v form an orthonormal basis if they are Each of them are unit length u = v = 1 Orthogonal to each other u. v = 0

27 Orthonormal bases In 3D: u, v and w form an orthonormal basis if they are All of unit length u = v = w = 1 Any pair of them are orthogonal to each other u. v = 0, v. w = 0, u. w = 0 Right handed if w = u v Otherwise len handed There are infinitely many orthonormal bases

28 Orthonormal bases In 3D: u, v and w form an orthonormal basis if they are All of unit length u = v = w = 1 Any pair of them are orthogonal to each other u. v = 0, v. w = 0, u. w = 0 Right handed if w = u v Otherwise len handed There are infinitely many orthonormal bases

29 Canonical orthonormal basis Cartesian canonical orthonormal basis x = (1, 0, 0) y = (0, 1, 0) z = (0, 0, 1)

30 Constructing an orthonormal basis given a single vector Easier on the board

31 Matrix What is a matrix IdenOty matrix Transpose Determinant Matrix vector muloplicaoon Matrix matrix muloplicaoon

π = d Addition Formulae Revision of some Trigonometry from Unit 1Exact Values H U1 O3 Trigonometry 26th November 2013 H U2 O3 Trig Formulae

π = d Addition Formulae Revision of some Trigonometry from Unit 1Exact Values H U1 O3 Trigonometry 26th November 2013 H U2 O3 Trig Formulae 26th November 2013 H U1 O3 Trigonometry Addition Formulae Revision of some Trigonometry from Unit 1Exact Values 2 30 o 30 o 3 2 r π = d 180 1 1 45 o 45 o 2 45 o 45 o 1 1 60 o 60 o 1 2 1 180 x x 180 + x